Recently, we proposed a symbolic method for algorithmic reduction of chemical reaction networks with multiple timescales [1]}. This method combines
tropical geometry ideas
for identifying the time and concentration scales
[2]}, [3]},
dominance principles based on comparison of
orders of magnitude [4]}, [5]}, [6]}
and
singular perturbation results [7]}, [8]} to justify the reduction.
For a given timescale, the method defines three subsystems: a slaved equilibrated subsystem, a driving evolving subsystem and a quenched
subsystem.
The variables of the slaved subsystem satisfy quasi-steady equations,
defined as equilibria of the fast truncated ODEs in which all the remaining variables are considered fixed.
The entire construction requires
the hyperbolicity of the quasi-steady state, which is needed in the classical geometrical
singular perturbation theory of Fenichel [7]}.
Although general in its implementation, this reduction method fails in a number of cases. A major cause of failure is the degeneracy of the quasi-steady state, when the fast dynamics
has a continuous variety of steady states. Typically, this happens when the fast truncated ODEs have first integrals,
i.e. quantities that are conserved on any trajectory and that have to be fixed at arbitrary values depending on the initial conditions. The quasi-steady states are no longer hyperbolic, because
the Jacobian matrix of the fast part of the dynamics is in this case singular.
This type of singular perturbations, called critical, is known since the work of Vasil'eva and Butuzov in the 70's [10]}, but its origins can be
traced back to the early theory of enzymatic reactions, as quasi-equilibrium is an
instance of critical singular perturbations. Vasil'eva and Butuzov
[10]} propose asymptotic expansions of the solutions of
singularly perturbed systems in the critical case
based on their method of boundary series, which are two timescale expansions.
Their method works in the case when the problem has rigorously two timescales, but
can not be applied in the case when there are more than two timescales.
We show in this paper that critical singular perturbations may have more timescales
than are apparent after rescaling parameters and variables. These extra timescales correspond to
slow approximate conserved quantities.
